System models. We look at LTI systems for the time being Time domain models

Transcription

1 Stem moel We look at LTI tem for the time being Time omain moel High orer orinar ifferential equation moel Contain onl input variable, output variable, their erivative, an contant parameter Proper: highet output erivative orer i greatet Highet orer erivative of output tem orer n t n a n b m m n t n a t m u b t u b 0u t a 0

2 Stem moel Time omain moel State pace moel: tate equation output equation State equation: a et of t orer iff eq on tate variable Output equation: output a function of tate an input! " Linear tem:! " f (,u g(,u A Bu C Du

3 u a t a t a t n n n n n 0 ODE moel to State pace moel Let,, 3,... Then, 3, 3 4,... t n! " $ % a 0 a a a n! " $ % n! " $ % 0 0! " $ % u [ ] [0]u

4 u z a z t a z t a z t z b z t b z t b u b u t b u t b a t a t a t n n n n n m m m m m m n n n n n Then : Let When : ODE moel to State pace moel m<n

5 ODE moel to State pace moel Let z, z, 3 z,... We till have the ame tate equation:! $! t " a n % " 0 a a a n But the output equation will be: m b m t z b m t z b z 0 b m m b b 0 [ b 0 b... b m ] [0]u $! % " n $! % " 0 0 $ u %

6 The eigenfunction φ k (t an their propertie (Focu on CT tem now, but reult appl to DT tem a well. eigenvalue eigenfunction Eigenfunction in ame function out with a gain From the uperpoition propert of LTI tem: Now the tak of fining repone of LTI tem i to etermine λ k. The olution i imple, general, an inightful. 6

7 that work for an an all Comple Eponential are the onl Eigenfunction of an LTI Stem eigenvalue eigenfunction eigenvalue eigenfunction

8 DT: 8

9 Tranfer Function 9

10 Tranfer Function Tranfer function from to i the gain from X( to Y(, that i, it i the ratio of Laplace tranform of to Laplace tranform of : [ ] Y ( X( L (t L (t [ ] Δ H(, or G(, Then: Y ( H(X(

11 Input Output Stem Input (t H( Output (t Y( H(X( If the input (t δ(t, the output i calle the impule repone. If the input (t u(t, the output i calle the tep repone. If the input (t Ain(ωt, an H( i table, output tea tate i A H(jω in(ωt H(jω Pole: value of at which TF à infinit Zero: value of at which TF 0

12 Eample: controller E( controller C( U( Proportional controller: C( K P cont Integral controller: C( K I / Derivative controller: C( K D PI controller: C( K P K I / PD controller: C( K P K D PID controller: C( K P K I / K D

13 State pace moel to TF ( ] ( [ ( ( ( ( ( ( ( ( ( ( ( ( ( : take State pace moel: ( D U B A SI C Y BU A SI X BU X A SI DU CX Y BU AX X Du C Bu A H L A, B, C, D are matrice

14 Block iagram computaon - CONTROLLER CONTROLLED DEVICE FEEDBACK ELEMENT

15

16 A line i a ignal A block i a gain A circle i a um Block Diagram Due to h.f. noie, ue proper block: num eg en eg Tr to ue jut horizontal or vertical line Ue aitional Σ to help e.g. - G Σ - z z G z -

17 Serie: Block Diagram Algebra G G è G G Parallel: G G è G G

18 Feeback: Proof: G G G G G - b e G G G G G e e G G G G e e G G G G e G b b e (,, è

19 G G G G G - n n n n n

20 >> tf('' Tranfer function: >> G(/( Tranfer function: >> G5/(5 Tranfer function: >> GG*G Tranfer function: ^ 7 0 >> HGG Tranfer function: ^ ^ 7 0 >> HFfeeback(G, G Tranfer function: ^ ^ 5

21 >> elatf(,,'inputela',0.05 Tranfer function: 0.9 ep(-0.05* * 0.8 Step Repone >> HHF*ela Tranfer function: ^ 6 5 ep(-0.05* * ^ 5 Amplitue >> tepreph*/ Tranfer function: ^ 6 5 ep(-0.05* * >> tep(h ^3 ^ Time (ec

22 Quarter car upenion Serie R( b k - m R( - b k m Feeback R( b k m b k TF b k H( m b k

23 >> bm('b'; >> mm('m'; >> km('k'; >> m(''; >> Gb*k G b*k >> G/m*/*/ G /m/^ >> GG*G G (b*k/m/^ >> GclG/(G Gcl (b*k/m/^/((b*k/m/^ >> implif(gcl an (b*k/(m*^b*k

24 Move a block (G acro a into all touching line: If arrow irection change, invert block (/G If arrow irection remain, no change in block For eample: pick-up point ummation along arrow along arrow no change G G G G z G 3 no change G along arrow along arrow z G 3

25 againt, againt z G G 3 G againt along è z G G 3 G /G G G è G G 3 /G 3 G z G 3 z

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28 I U - I - L R C V c L R R I L R U - - L R C V c L R R U C ( L R L R L R - R -

29 U L R C( L R L R - R L R U C ( L L R - R R ( L R U C ( L L R L R ( L R R R T. F. C( L R ( L R R L R L R

30 Fin TF from U to Y: 5 U ( 0 Y No pure erie/parallel/feeback Nee to move a block, but which one? Ke: move one block to create pure erie or parallel or feeback! 0 So move either left or right. ( 0

31 ( U ( 0 Y U ( ( ( 0 ( 0 5( 5 Y U ( ( ( 0 0( ( 0 5( 5 Y

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33 Can ue uperpoition: Firt et D0, fin Y ue to R Then et R0, fin Y ue to D Finall, a the two component to get the overall Y

34 Firt et D0, fin Y ue to R Y GG ( R( GG H

35 Then et R0, fin Y ue to D G Y ( D( ( GG H G

36 Finall, a the two component to get the overall Y GG G Y R D GG H GG H ( ( (